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calcmogul / controls-engineering-in-frc

Controls Engineering in the FIRST Robotics Competition: Graduate-level control theory for high schoolers.
BSD 3-Clause "New" or "Revised" License
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Derivative in discrete version of state-space definition #8

Closed WasabiFan closed 5 years ago

WasabiFan commented 5 years ago

Section 5.3.3, Definition 5.3.1, shows a continuous and discrete version of the same equation: image

In the continuous case, the LHS is the derivative of x, but in the discrete case the derivative is dropped and no delta takes its place: the LHS simply reads x_{k+1}. Why is this?

calcmogul commented 5 years ago

Although it isn't clear from the notation, the continuous and discrete versions have different A and B matrices. The discrete equation returns the next state rather than a change in state because while you could have x_{k+1} - x_k = Ax_k + Bu_k, you have to do an extra addition by x_k to propagate the model forward. Also note that the poles would be given by the eigenvalues of A + I rather than the eigenvalues of A. I know a mathematics guy that's annoyed by the deviation in notation from the continuous version, but :man_shrugging:. Practicality beats purity in this case.

Section 6.5 "Matrix exponential" through 6.7 go into more detail on what the discrete matrices are defined as, and G.3 "Zero-order hold for state-space" shows a derivation.

WasabiFan commented 5 years ago

Thanks for the clarification! I appreciate the quick response. It might be good to have a note about that near the definition.